Can we deduce the correct Calabi-Yau space?

As I understand it, String theory requires 6 other dimensions compactified so small we can't measure them directly. But the way they compactify determines the physical constants of the universe such as the mass of the electron and the strength of gravity and the EM field, etc. Yet there is a plethora of ways in which the extra dimensions could compactify, and we don't know how to uniquely determine which one fits our observations. OK. My question is can we work back-wards from the constants and determine which way the Calabi-Yau must be compactified?

Many such "heterotic standard models" are now known. But at low energy they all produce, not just a standard model (though not with the masses and couplings of the real world), but a standard model plus supersymmetry (the "MSSM", "minimal supersymmetric standard model"). I think the hope of many would be that the LHC will reveal details of the supersymmetric sector, providing new qualitative guidance in the search for the right compactification.

Actually calculating the masses and couplings predicted by any of these heterotic standard models is also a huge challenge. The masses come from "yukawas", the couplings of fermions with the Higgs, and there is a paper from 2006 in which a first approximation to the yukawas in that first heterotic standard model are calculated. Their grand deduction is that one generation of fermions will be massless, and the other two massive with unequal masses. But that's just the first approximation, and there will be additional effects that modify the yukawas (e.g. so that the "massless" generation acquires light masses); also, the exact values of the yukawas depend on where the minima of several complicated geometric functions are located.

That paper from 2006 has had a few follow-ups, but in general, I don't see people computing yukawas - they are still refining their understanding of other features of such compactifications. Also let us remember that the MSSM may not be the right low-energy theory! It may instead be one of the many modifications now being considered by phenomenologists; or perhaps supersymmetry is entirely absent at low energies.

As for working in the other direction - starting with the masses, etc, and working towards the deeper theory - this is more the province of grand unified theories. People will use elementary qualitative features of the masses and mixings (e.g. "one quark - the top quark - is much heavier than all the others"), in order to infer the possible structure of a grand unified theory. Neutrinos are a popular source of such speculation right now - dozens of papers are being written, in which new symmetries are introduced, in order to explain the measured neutrino masses and mixings. Any such grand unified theory (e.g. "SO(10) gauge symmetry with discrete A4 symmetry") is potentially an improved target for string phenomenologists to aim at.

So people are working from the bottom up and from the top down, as you surmise, but it's a rather involved process.